Maximal Functions in Sobolev Spaces - Aalto Math

Maximal Functions in Sobolev Spaces Daniel Aalto and Juha Kinnunen

Abstract Applications of the Hardy–Littlewood maximal functions in the modern theory of partial differential equations are considered. In particular, we discuss the behavior of maximal functions in Sobolev spaces, Hardy inequalities, and approximation and pointwise behavior of Sobolev functions. We also study the corresponding questions on metric measure spaces.

1 Introduction The centered Hardy–Littlewood maximal function M f : Rn → [0, ∞] of a locally integrable function f : Rn → [−∞, ∞] is defined by Z |f (y)| dy, M f (x) = sup B(x,r)

where the supremum is taken over all radii r > 0. Here Z Z 1 |f (y)| dy = |f (y)| dy |B(x, r)| B(x,r)

B(x,r)

denotes the integral average and |B(x, r)| is the volume of the ball B(x, r). There are several variations of the definition in the literature, for example, Daniel Aalto Institute of Mathematics, P.O. Box 1100, FI-02015 Helsinki University of Technology, Finland, e-mail: [email protected] Juha Kinnunen Institute of Mathematics, P.O. Box 1100, FI-02015 Helsinki University of Technology, Finland, e-mail: [email protected]

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Daniel Aalto and Juha Kinnunen

depending on the requirement whether x is at the center of the ball or not. These definitions give maximal functions that are equivalent with two-sided estimates. The maximal function theorem of Hardy, Littlewood, and Wiener asserts that the maximal operator is bounded in Lp (Rn ) for 1 < p 6 ∞, kM f kp 6 ckf kp ,

(1.1)

where c = c(n, p) is a constant. The case p = ∞ follows immediately from the definition of the maximal function. It can be shown that for the centered maximal function the constant depends only on p, but we do not need this fact here. For p = 1 we have the weak type estimate |{x ∈ Rn : M f (x) > λ}| 6 cλ−1 kf k1 for every λ > 0 with c = c(n) (see [64]). The maximal functions are classical tools in harmonic analysis. They are usually used to estimate absolute size, and their connections to regularity properties are often neglected. The purpose of this exposition is to focus on this issue. Indeed, applications to Sobolev functions and to partial differential equations indicate that it is useful to know how the maximal operator preserves the smoothness of functions. There are two competing phenomena in the definition of the maximal function. The integral average is smoothing but the supremum seems to reduce the smoothness. The maximal function is always lower semicontinuous and preserves the continuity of the function provided that the maximal function is not identically infinity. In fact, if the maximal function is finite at one point, then it is finite almost everywhere. A result of Coifman and Rochberg states that the maximal function raised to a power which is strictly between zero and one is a Muckenhoupt weight. This is a clear evidence of the fact that the maximal operator may have somewhat unexpected smoothness properties. It is easy to show that the maximal function of a Lipschitz function is again Lipschitz and hence, by the Rademacher theorem is differentiable almost everywhere. The question about differentiability in general is a more delicate one. Simple one-dimensional examples show that the maximal function of a differentiable function is not differentiable in general. Nevertheless, certain weak differentiability properties are preserved under the maximal operator. Indeed, the Hardy–Littlewood maximal operator preserves the first order Sobolev spaces W 1,p (Rn ) with 1 < p 6 ∞, and hence it can be used as a test function in the theory of partial differential equations. More precisely, the maximal operator is bounded in the Sobolev space and for every 1 < p 6 ∞ we have kM uk1,p 6 ckuk1,p with c = c(n, p). We discuss different aspects related to this result.

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The maximal functions can also be used to study the smoothness of the original function. Indeed, there are pointwise estimates for the function in terms of the maximal function of the gradient. If u ∈ W 1,p (Rn ), 1 6 p 6 ∞, then there is a set E of measure zero such that ¡ ¢ |u(x) − u(y)| 6 c|x − y| M |Du|(x) + M |Du|(y) for all x, y ∈ Rn \E. If 1 < p 6 ∞, then the maximal function theorem implies that M |Du| ∈ Lp (Rn ). This observation has fundamental consequences in the theory of partial differential equations. Roughly speaking, the oscillation of the function is small on the good set where the maximal function of the gradient is bounded. The size of the bad set can be estimated by the maximal function theorem. This can also be used to define Sobolev type spaces in a very general context of metric measure spaces. To show that our arguments are based on a general principle, we also consider the smoothness of the maximal function in this case. The results can be used to study the pointwise behavior of Sobolev functions.

2 Maximal Function Defined on the Whole Space Recall that the Sobolev space W 1,p (Rn ), 1 6 p 6 ∞, consists of functions u ∈ Lp (Rn ) whose weak first order partial derivatives Di u, i = 1, 2, . . . , n, belong to Lp (Rn ). We endow W 1,p (Rn ) with the norm kuk1,p = kukp + kDukp , where Du = (D1 u, D2 u, . . . , Dn u) is the weak gradient of u. Equivalently, if 1 6 p < ∞, the Sobolev space can be defined as the completion of smooth functions with respect to the norm above. For basic properties of Sobolev functions we refer to [17].

2.1 Boundedness in Sobolev spaces Suppose that u is Lipschitz continuous with constant L, i.e., |uh (y) − u(y)| = |u(y + h) − u(y)| 6 L|h| for all y, h ∈ Rn , where uh (y) = u(y + h). Since the maximal function commutes with translations and the maximal operator is sublinear, we have

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|(M u)h (x) − M u(x)| = |M (uh )(x) − M u(x)| 6 M (uh − u)(x) Z 1 = sup |uh (y) − u(y)| dy 6 L|h|. r>0 |B(x, r)|

(2.1)

B(x,r)

This means that the maximal function is Lipschitz continuous with the same constant as the original function provided that M u is not identically infinity [19]. Observe that this proof applies to H¨older continuous functions as well [14]. It is shown in [33] that the Hardy–Littlewood maximal operator is bounded in the Sobolev space W 1,p (Rn ) for 1 < p 6 ∞ and hence, in that case, it has classical partial derivatives almost everywhere. Indeed, there is a simple proof based on the characterization of W 1,p (Rn ) with 1 < p < ∞ by integrated difference quotients according to which u ∈ Lp (Rn ) belongs to W 1,p (Rn ) if and only if there is a constant c for which kuh − ukp 6 ckDukp |h| for every h ∈ Rn . As in (2.1), we have |M (uh ) − M u| 6 M (uh − u) and, by the Hardy–Littlewood–Wiener maximal function theorem, we conclude that k(M u)h − M ukp = kM (uh ) − M ukp 6 kM (uh − u)kp 6 ckuh − ukp 6 ckDukp |h| for every h ∈ Rn , from which the claim follows. A more careful analysis gives even a pointwise estimate for the partial derivatives. The following simple proposition is used several times in the sequel. If fj → f and gj → g weakly in Lp (Ω) and fj (x) 6 gj (x), j = 1, 2, . . . , almost everywhere in Ω, then f (x) 6 g(x) almost everywhere in Ω. Together with some basic properties of the first order Sobolev spaces, this implies that the maximal function semicommutes with weak derivatives. This is the content of the following result which was first proved in [33], but we recall the simple argument here (see also [40, 41]). Theorem 2.2. Let 1 < p < ∞. If u ∈ W 1,p (Rn ), then M u ∈ W 1,p (Rn ) and |Di M u| 6 M Di u,

i = 1, 2, . . . , n,

almost everywhere in Rn . Proof. If χB(0,r) is the characteristic function of B(0, r) and

(2.3)

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χr = then

χB(0,r) , |B(0, r)|

Z

1 |B(x, r)|

|u(y)| dy = |u| ∗ χr (x), B(x,r)

where ∗ denotes convolution. Now |u| ∗ χr ∈ W 1,p (Rn ) and Di (|u| ∗ χr ) = χr ∗ Di |u|,

i = 1, 2, . . . , n,

almost everywhere in Rn . Let rj , j = 1, 2, . . . , be an enumeration of the positive rational numbers. Since u is locally integrable, we may restrict ourselves to positive rational radii in the definition of the maximal function. Hence M u(x) = sup(|u| ∗ χrj )(x). j

We define functions vk : Rn → R, k = 1, 2, . . . , by vk (x) = max (|u| ∗ χrj )(x). 16j6k

Now (vk ) is an increasing sequence of functions in W 1,p (Rn ) which converges to M u pointwise and |Di vk | 6 max |Di (|u| ∗ χrj )| = max |χrj ∗ Di |u|| 16j6k

16j6k

6 M Di |u| = M Di u, i = 1, 2, . . . , n, almost everywhere in Rn . Here we also used the fact that |Di |u|| = |Di u|, i = 1, 2, . . . , n, almost everywhere. Thus, kDvk kp 6

n X

kDi vk kp 6

i=1

n X

kM Di ukp

i=1

and the maximal function theorem implies kvk k1,p 6 kM ukp +

n X

kM Di ukp

i=1

6 ckukp + c

n X

kDi ukp 6 c < ∞

i=1

for every k = 1, 2, . . . . Hence (vk ) is a bounded sequence in W 1,p (Rn ) which converges to M u pointwise. By the weak compactness of Sobolev spaces,

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M u ∈ W 1,p (Rn ), vk converges to M u weakly in Lp (Rn ), and Di vk converges to Di M u weakly in Lp (Rn ). Since |Di vk | 6 M Di u almost everywhere, the weak convergence implies |Di M u| 6 M Di u,

i = 1, 2, . . . , n,

almost everywhere in Rn .

u t

Remark 2.4. (i) The case p = 1 is excluded in the theorem because our arguments fail in that case. However, Tanaka [66] proved, in the one-dimensional case, that if u ∈ W 1,1 (R), then the noncentered maximal function is differentiable almost everywhere and kDM uk1 6 2kDuk1 . For extensions of Tanaka’s result to functions of bounded variation in the onedimensional case we refer to [3] and [4]. The question about the counterpart of Tanaka’s result remains open in higher dimensions (see also discussion in [26]). Observe that kukn/n−1 6 ckDuk1 by the Sobolev embedding theorem and M u ∈ Ln/(n−1) (Rn ) by the maximal function theorem. However, the behavior of the derivatives is not well understood in this case. (ii) The inequality (2.3) implies that |DM u(x)| 6 M |Du|(x)

(2.5)

for almost all x ∈ Rn . Fix a point at which the gradient DM u(x) exists. If |DM u(x)| = 0, then the claim is obvious. Hence we may assume that |DM u(x)| 6= 0. Let DM u(x) e= . |DM u(x)| Rotating the coordinates in the proof of the theorem so that e coincides with some of the coordinate directions, we get |DM u(x)| = |De M u(x)| 6 M Dh u(x) 6 M |Du|(x), where De u = Du · e is the derivative to the direction of the unit vector e. (iii) Using the maximal function theorem together with (2.3), we find kM uk1,p = kM ukp + kDM ukp 6 ckukp + kM |Du|kp 6 ckuk1,p , where c is the constant in (1.1). Hence

(2.6)

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M : W 1,p (Rn ) → W 1,p (Rn ) is a bounded operator, where 1 < p < ∞. (iv) If u ∈ W 1,∞ (Rn ), then a slight modification of our proof shows that M u belongs to W 1,∞ (Rn ). Moreover, kM uk1,∞ = kM uk∞ + kDM uk∞ 6 kuk∞ + kM |Du|k∞ 6 kuk1,∞ . Hence, in this case, the maximal operator is bounded with constant one. Recall that, after a redefinition on a set of measure zero, u ∈ W 1,∞ (Rn ) is a bounded and Lipschitz continuous function. (v) A recent result of Luiro [53] shows that M : W 1,p (Rn ) → W 1,p (Rn ) is a continuous operator. Observe that bounded nonlinear operators are not continuous in general. Luiro employs the structure of the maximal operator. He also obtained an interesting formula for the weak derivatives of the maximal function. Indeed, if u ∈ W 1,p (R), 1 < p < ∞, and R(x) denotes the set of radii r > 0 for which Z M u(x) = lim sup |u| dy ri →r

B(x,ri )

for some sequence (ri ) with ri > 0, then for almost all x ∈ Rn we have Z Di M u(x) = Di |u| dy B(x,r)

for every strictly positive r ∈ R(x) and Di M u(x) = Di |u|(x) if 0 ∈ R(x). For this is a sharpening of (2.3) we refer to [53, Theorem 3.1] (see also [55]). (vi) Let 0 6 α 6 n. The fractional maximal function of a locally integrable function f : Rn → [−∞, ∞] is defined by Z |f (y)| dy. Mα f (x) = sup rα r>0

B(x,r)

For α = 0 we obtain the Hardy–Littlewood maximal function.

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Theorem 2.2 can be easily extended to fractional maximal functions. Indeed, suppose that 1 < p < ∞. Let 0 6 α < n/p. If u ∈ W 1,p (Rn ), then Mα u ∈ W 1,q (Rn ) with q = np/(n − αp) and |Di Mα u| 6 Mα Di u,

i = 1, 2, . . . , n,

almost everywhere in Rn . Moreover, there is c = c(n, p, α) such that kMα uk1,q 6 c kuk1,p . The main result of [39] shows that the fractional maximal operator is smoothing in the sense that it maps Lp -spaces into certain first order Sobolev spaces.

2.2 A capacitary weak type estimate As an application, we show that a weak type inequality for the Sobolev capacity follows immediately from Theorem 2.2. The standard proofs seem to depend, for example, on certain extension properties of Sobolev functions (see [17]). Let 1 < p < ∞. The Sobolev p-capacity of the set E ⊂ Rn is defined by Z ¡ p ¢ |u| + |Du|p dx, capp (E) = inf u∈A(E) Rn

where © ª A(E) = u ∈ W 1,p (Rn ) : u > 1 on a neighborhood of E . If A(E) = ∅, we set capp (E) = ∞. The Sobolev p-capacity is a monotone and countably subadditive set function. Let u ∈ W 1,p (Rn ). Suppose that λ > 0 and denote Eλ = {x ∈ Rn : M u(x) > λ}. Then Eλ is open and M u/λ ∈ A(Eλ ). Using (2.6), we get ¡

capp Eλ

¢

1 6 p λ

6

c λp

Z

¡ ¢ |M u|p + |DM u|p dx

Rn

Z Rn

¡ p ¢ c |u| + |Du|p dx 6 p kukp1,p . λ

This inequality can be used in the study of the pointwise behavior of Sobolev functions by standard methods. We recall that x ∈ Rn is a Lebesgue point for u if the limit

Maximal Functions in Sobolev Spaces

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Z u∗ (x) = lim

u dy

r→0 B(x,r)

exists and

Z |u(y) − u∗ (x)| dy = 0.

lim

r→0 B(x,r)

The Lebesgue theorem states that almost all points of a L1loc (Rn ) function are Lebesgue points. If a function belongs to W 1,p (Rn ), then, using the capacitary weak type estimate, we can prove that the complement of the set of Lebesgue points has zero p-capacity (see [17]).

3 Maximal Function Defined on a Subdomain Let Ω be an open set in the Euclidean space Rn . For a locally integrable function f : Ω → [−∞, ∞] we define the Hardy–Littlewood maximal function MΩ f : Ω → [0, ∞] as Z MΩ f (x) = sup |f (y)| dy, B(x,r)

where the supremum is taken over all radii 0 < r < δ(x), where δ(x) = dist(x, ∂Ω). In this section, we make the standing assumption that Ω 6= Rn so that δ(x) is finite. Observe that the maximal function depends on Ω. The maximal function theorem implies that the maximal operator is bounded in Lp (Ω) for 1 < p 6 ∞, i.e., kMΩ f kp,Ω 6 ckf kp,Ω . (3.1) This follows directly from (1.1) by considering the zero extension to the complement. The Sobolev space W 1,p (Ω), 1 6 p 6 ∞, consists of those functions u which, together with their weak first order partial derivatives Du = (D1 u, . . . , Dn u), belong to Lp (Ω). When 1 6 p < ∞, we may define W 1,p (Ω) as the completion of smooth functions with respect to the Sobolev norm.

3.1 Boundedness in Sobolev spaces We consider the counterpart of Theorem 2.2 for the maximal operator MΩ . It turns out that the arguments in the previous section do not apply mainly

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because the maximal operator MΩ does not commute with translations. The following result was proved in [35]. We also refer to [26] for an alternative approach. Theorem 3.2. Let 1 < p 6 ∞. If u ∈ W 1,p (Ω), then MΩ u ∈ W 1,p (Ω) and |DMΩ u| 6 2MΩ |Du| almost everywhere in Ω. Observe that the result holds for every open set and, in particular, we do not make any regularity assumption on the boundary. The functions ut : Ω → [−∞, ∞], 0 < t < 1, defined by Z ut (x) = |u(y)| dy, B(x,tδ(x))

will play a crucial role in the proof of Theorem 3.2 because MΩ u(x) = sup ut (x) 0 0. Then we show that (uλ ) is a uniformly bounded family of functions in W01,p (Ω). Clearly, uλ 6 |u| and hence Z Z p uλ dx 6 |u|p dx. Ω

Ω

For the gradient estimate we define Fλ = {x ∈ Ω : |u(x)| > λ dist(x, ∂Ω)}, where λ > 0. Then Z Z p |Duλ | dx = Ω

Z p

|Du| dx + λ

p

|D dist(x, ∂Ω)|p dx Fλ

Ω\Fλ

Z

|Du|p dx + λp |Fλ |,

6 Ω

where, by assumption, Z µ p

λ |Fλ | 6

|u(x)| dist(x, ∂Ω)

¶p dx < ∞

Ω

for every λ > 0. Here we again used the fact that |D dist(x, ∂Ω)| = 1 for almost all x ∈ Ω. This implies that (uλ ) is a uniformly bounded family of functions in W01,p (Ω). Since |Fλ | → 0 as λ → ∞ and uλ = |u| in Ω \ Fλ , we have uλ → |u| almost everywhere in Ω. A similar weak compactness argument that was used in the proofs of Theorems 2.2 and 3.2 shows that |u| ∈ W01,p (Ω). u t Remark 3.11. The proof shows that, instead of u/δ ∈ Lp (Ω), it is enough to assume that u/δ belongs to the weak Lp (Ω). Boundary behavior of the maximal function was studied in [37, 35] Theorem 3.12. Let Ω ⊂ Rn be an open set. Suppose that u ∈ W 1,p (Ω) with p > 1. Then

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|u| − MΩ u ∈ W01,p (Ω). Remark 3.13. In particular, if u ∈ W01,p (Ω), then MΩ u ∈ W01,p (Ω). Observe that this holds for every open subset Ω. Proof. Fix 0 < t < 1. A standard telescoping argument (see Lemma 4.1) gives ¯ ¯ ¯ ¯ Z ¯ ¯ ¯ ¯ ¯ ¯ ¯|u(x)| − ut (x)¯ = ¯|u(x)| − |u(y)| dy ¯ ¯ ¯ ¯ ¯ B(x,tδ(x)) 6 ct dist(x, ∂Ω)MΩ |Du|(x). For every x ∈ Ω there is a sequence tj , j = 1, 2, . . . , such that MΩ u(x) = lim utj (x). j→∞

This implies that ¯ ¯ ¯ ¯ ¯|u(x)| − MΩ u(x)¯ = lim ¯|u(x)| − ut (x)¯ j j→∞

6 c dist(x, ∂Ω)MΩ |Du|(x). By the maximal function theorem, we conclude that ¯ !p Z Z à ¯¯ |u(x)| − MΩ u(x)¯ dx 6 c (MΩ |Du|(x))p dx dist(x, ∂Ω) Ω

Ω

Z |Du(x)|p dx.

6c Ω

This implies that

|u(x)| − MΩ u(x) ∈ Lp (Ω). dist(x, ∂Ω)

By Theorem 3.2, we have MΩ u ∈ W 1,p (Ω), and from Lemma 3.10 we conclude that |u| − MΩ u ∈ W01,p (Ω). u t Remark 3.14. We observe that the maximal operator preserves nonnegative superharmonic functions; see [37]. (For superharmonic functions that change signs, we may consider the maximal function without absolute values.) Suppose that u : Ω → [0, ∞] is a measurable function which is not identically ∞ on any component of Ω. Then it is easy to show that MΩ u(x) = u(x)

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41

for every x ∈ Ω if and only if u is superharmonic. The least superharmonic majorant can be constructed by iterating the maximal function. For short we write (k)

MΩ u(x) = MΩ ◦ MΩ ◦ · · · ◦ MΩ u(x),

k = 1, 2, . . . .

(k)

Since MΩ u, k = 1, 2, . . . , are lower semicontinuous, we see that (k)

(k+1)

MΩ u(x) 6 MΩ

u(x),

k = 1, 2, . . . ,

(k)

for every x ∈ Ω. Hence (MΩ u(x)) is an increasing sequence of functions and it converges for every x ∈ Ω (the limit may be ∞). We denote (∞)

(k)

MΩ u(x) = lim MΩ u(x) k→∞

(∞)

for every x ∈ Ω. If MΩ u is not identically infinity on any component of Ω, then it is the smallest superharmonic function with the property that (∞)

MΩ u(x) > u(x) for almost all x ∈ Ω. If u ∈ W 1,p (Ω), then the obtained smallest superharmonic function has the same boundary values as u in the Sobolev sense by Theorem 3.12. Fiorenza [18] observed that nonnegative functions of one or two variables cannot be invariant under the maximal operator unless they are constant. This is consistent with the fact that on the line there are no other concave functions and in the plane there are no other superharmonic functions but constants that are bounded from below (see also [42]).

4 Pointwise Inequalities The following estimates are based on a well-known telescoping argument (see [28] and [16]). The proofs are based on a general principle and they apply in a metric measure space equipped with a doubling measure (see [25]). This fact will be useful below. Let 0 < β < ∞ and R > 0. The fractional sharp maximal function of a locally integrable function f is defined by Z # fβ,R (x) = sup r−β |f − fB(x,r) | dy, 0 0, such that the Poincar´e inequality holds, Z Z |u − uB(x,r) | dy 6 c r g dy B(x,r)

B(x,r)

for all x ∈ Rn and r > 0. p n (iv) u ∈ Lp (Rn ) and u# 1 ∈ L (R ).

Proof. We have already seen that (i) implies (ii). To prove that (ii) implies (iii), we integrate the pointwise inequality twice over the ball B(x, r). After the first integration we obtain ¯ ¯ ¯ ¯ Z ¯ ¯ ¯ ¯ |u(y) − uB(x,r) | = ¯u(y) − u(z) dz ¯ ¯ ¯ ¯ ¯ B(x,r) Z |u(y) − u(z)| dz

6 B(x,r)

  6 2r g(y) +

Z

  g(z) dz  ,

B(x,r)

which implies Z



Z

 |u(y) − uB(x,r) | dy 6 2r 

B(x,r)

Z g(y) dy +

B(x,r)

Z 6 4r

g(y) dy. B(x,r)

To show that (iii) implies (iv), we observe that

B(x,r)

  g(z) dz 

Maximal Functions in Sobolev Spaces

u# 1 (x) = sup r>0

1 r

45

Z

Z |u − uB(x,r) | dy 6 c sup

g dy = cM g(x).

r>0

B(x,r)

B(x,r)

Then we show that (iv) implies (i). By Theorem 4.1, # |u(x) − u(y)| 6 c|x − y|(u# 1 (x) + u1 (y)) p n for all x, y ∈ Rn \ E with |E| = 0. If we denote g = cu# 1 , then g ∈ L (R ) and |u(x) − u(y)| 6 |x − y|(g(x) + g(y))

for all x, y ∈ Rn \ E with |E| = 0. Then we use the characterization of Sobolev spaces W 1,p (Rn ), 1 < p < ∞, with integrated difference quotients. Let h ∈ Rn . Then |uh (x) − u(x)| = |u(x + h) − u(x)| 6 |h|(gh (x) + g(x)), from which we conclude that kuh − ukp 6 |h|(kgh kp + kgkp ) = 2|h|kgkp , which implies the claim.

u t

Remark 4.5. HajÃlasz [22] showed that u ∈ W 1,1 (Rn ) if and only if u ∈ L1 (Rn ) and there is a nonnegative function g ∈ L1 (Rn ) and σ > 1 such that |u(x) − u(y)| 6 |x − y|(Mσ|x−y| g(x) + Mσ|x−y| g(y)) for all x, y ∈ Rn \ E with |E| = 0. Moreover, if this inequality holds, then |Du| 6 c(n, σ)g almost everywhere.

4.1 Lusin type approximation of Sobolev functions Approximations of Sobolev functions were studied, for example, in [2, 10, 11, 13, 20, 25, 52, 56, 58, 60, 69]. Let u ∈ W 1,p (Rn ) and 0 6 α < 1. By Corollary (4.3), ¡ ¢ |u(x) − u(y)| 6 c|x − y|1−α Mα |Du|(x) + Mα |Du|(y) for all x, y ∈ Rn \ E with |E| = 0. For p > n the H¨older inequality implies Mn/p |Du|(x) 6 cMn |Du|p (x)1/p 6 ckDukp for every x ∈ Rn \ E with c = c(n, p). Hence

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|u(x) − u(y)| 6 ckDukp |x − y|1−n/p for all x, y ∈ Rn \ E and u is H¨older continuous with the exponent 1 − n/p after a possible redefinition on a set of measure zero. The same argument implies that if Mα |Du| is bounded, then u ∈ C 1−α (Rn ). Even if Mα |Du| is unbounded, then |u(x) − u(y)| 6 cλ|x − y|1−α for all x, y ∈ Rn \ Eλ , where Eλ = {x ∈ Rn : Mα |Du|(x) > λ} for λ > 0. This means that the restriction of u ∈ W 1,p (Rn ) to the set Rn \Eλ is H¨older continuous after a redefinition on a set of measure zero. Recall that the (spherical) Hausdorff s-content, 0 < s < ∞, of E ⊂ Rn is defined by ∞ ∞ nX o [ s H∞ (E) = inf ris : E ⊂ B(xi , ri ) . i=1

i=1

The standard Vitali covering argument gives the following estimate for the size of the set Rn \ Eλ . There is a constant c = c(n, p, α) such that Z n−αp −p H∞ (Eλ ) 6 cλ |Du|p dx (4.6) Rn

for every λ > 0. Theorem 4.7. Let u ∈ W 1,p (Rn ), and let 0 6 α < 1. Then for every λ > 0 there is an open set Eλ and a function uλ such that u(x) = uλ (x) for every x ∈ Rn \ Eλ , uλ ∈ W 1,p (Rn ), uλ is H¨ older continuous with the exponent n−αp 1 − α, ku − uλ kW 1,p (Rn ) → 0 as λ → ∞, and H∞ (Eλ ) → 0 as λ → ∞. Remark 4.8. (i) If α = 0, then the theorem says that every function in the Sobolev space coincides with a Lipschitz function outside a set of arbitrarily small Lebesgue measure. The obtained Lipschitz function approximates the original Sobolev function also in the Sobolev norm. (ii) Since

n−αp capαp (Eλ ) 6 cH∞ (Eλ ),

the size of the exceptional set can also be expressed in terms of capacity. Proof. The set Eλ is open since Mα is lower semicontinuous. From (4.6) we conclude that n−αp H∞ (Eλ ) 6 cλ−p kDukpp for every λ > 0 with c = c(n, p, α). We already showed that u|Rn \Eλ is (1 − α)-H¨older continuous with the constant c(n)λ.

Maximal Functions in Sobolev Spaces

47

Let Qi , i = 1, 2, . . ., be a Whitney decomposition of Eλ with the following properties: each Qi is open, the cubes Qi , i = 1, 2, . . ., are disjoint, Eλ = S∞ i=1 Qi , 4Qi ⊂ Eλ , i = 1, 2, . . ., ∞ X

χ2Qi 6 N < ∞,

i=1

and

c1 dist(Qi , Rn \ Eλ ) 6 diam(Qi ) 6 c2 dist(Qi , Rn \ Eλ )

for some constants c1 and c2 . Then we construct a partition of unity associated with the covering 2Qi , i = 1, 2, . . . . This can be done in two steps. First, let ϕ ei ∈ C0∞ (2Qi ) be such that 0 6 ϕ ei 6 1, ϕ ei = 1 in Qi and |Dϕ ei | 6

c diam(Qi )

for i = 1, 2, . . . . Then we define ϕ ei (x) ϕi (x) = P ∞ ϕ ej (x) j=1

for every i = 1, 2, . . .. Observe that the sum is taken over finitely many terms only since ϕi ∈ C0∞ (2Qi ) and the cubes 2Qi , i = 1, 2, . . ., are of bounded overlap. The functions ϕi have the property ∞ X

ϕi (x) = χEλ (x)

i=1

for every x ∈ Rn . Then we define the function uλ by  u(x), ∞ uλ (x) = P  ϕi (x)u2Qi , i=1

x ∈ Rn \ Eλ , x ∈ Eλ .

The function uλ is a Whitney type extension of u|Rn \Eλ to the set Eλ . First we claim that kuλ kW 1,p (Eλ ) 6 ckukW 1,p (Eλ ) . Since the cubes 2Qi , i = 1, 2, . . ., are of bounded overlap, we have

(4.9)

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Daniel Aalto and Juha Kinnunen

Z

Z ¯X ∞ ∞ Z ¯p X ¯ ¯ p |uλ | dx = ϕi (x)u2Qi ¯ dx 6 c |u2Qi |p dx ¯

Eλ

i=1

Eλ

6c

∞ X

i=12Q

Z

Z |u|p dx 6 c

|2Qi |

i=1

i

2Qi

|u|p dx.

Eλ

Then we estimate the gradient. We recall that Φ(x) =

∞ X

ϕi (x) = 1

i=1

for every x ∈ Eλ . Since the cubes 2Qi , i = 1, 2, . . ., are of bounded overlap, we see that Φ ∈ C ∞ (Eλ ) and ∞ X

Dj Φ(x) =

Dj ϕi (x) = 0,

j = 1, 2, . . . , n,

i=1

for every x ∈ Eλ . Hence we obtain ∞ ∞ ¯X ¯ ¯X ¯ ¯ ¯ ¯ ¯ |Dj uλ (x)| = ¯ Dj ϕi (x)u2Qi ¯ = ¯ Dj ϕi (x)(u(x) − u2Qi )¯ i=1

6c

∞ X

i=1

diam(Qi )−1 |u(x) − u2Qi |χ2Qi (x)

i=1

and, consequently, ∞ X

|Dj uλ (x)| 6 c

diam(Qi )−p |u(x) − u2Qi |p χ2Qi (x).

i=1

Here we again used the fact that the cubes 2Qi , i = 1, 2, . . ., are of bounded overlap. This implies that for every j = 1, 2, . . . , n Z |Dj uλ | dx 6 c Eλ

Z ³X ∞

Eλ

6

i=1

∞ Z X i=12Q

´ diam(Qi )−p |u − u2Qi |p χ2Qi dx

diam(Qi )−p |u − u2Qi |p dx i

Maximal Functions in Sobolev Spaces

6c

∞ Z X

49

Z |Du|p dx 6 c

i=12Q

Eλ

i

1,p

|Du|p dx.

n

Then we show that uλ ∈ W (R ). We know that uλ belongs to W 1,p (Eλ ) and is H¨older continuous in Rn . Moreover, u ∈ W 1,p (Rn ) and u = uλ in Rn \ Eλ by (i). This implies that w = u − uλ ∈ W 1,p (Eλ ) and w = 0 in Rn \ Eλ . By the ACL-property, u is absolutely continuous on almost every line segment parallel to the coordinate axes. Take any such a line. Now w is absolutely continuous on the part of the line segment which intersects Eλ . On the other hand, w = 0 in the complement of Eλ . Hence the continuity of w in the line segment implies that w is absolutely continuous on the whole line segment. We have ku − uλ kW 1,p (Rn ) = ku − uλ kW 1,p (Eλ ) 6 kukW 1,p (Eλ ) + kuλ kW 1,p (Eλ ) 6 ckukW 1,p (Eλ ) . We leave it as an exercise for the interested reader to show that the function uλ is H¨older continuous with the exponent 1 − α (or see, for example, [27] for details). u t

5 Hardy Inequality In this section, we consider the Hardy inequality, which was originally studied by Hardy in the one-dimensional case. In the higher dimensional case, the Hardy inequality was studied, for example, in [5, 46, 51, 59, 67, 68]. Our approach is mainly based on more recent works [23, 36, 44, 47, 48, 61]. Suppose first that p > n, n < q < p, 0 6 α < q, and Ω 6= Rn is an open set. Let u ∈ C0∞ (Ω). Consider the zero extension to Rn \ Ω. Fix x ∈ Ω and take x0 ∈ ∂Ω such that |x − x0 | = dist(x, ∂Ω) = δ(x) = R. Denote χ = χB(x0 ,2R) . By Corollary 4.3, |u(x)| = |u(x) − u(x0 )| 6 c|x − x0 |1−n/q (Mn/q (|Du|χ)(x) + Mn/q (|Du|χ)(x0 )), where

Mn/q (|Du|χ)(x) 6 cMn (|Du|q χ)(x)1/q 6 kDuχkq ,

and, by the same argument,

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Daniel Aalto and Juha Kinnunen

Mn/q (|Du|χ)(x0 ) 6 kDuχkq . This implies that 

1/q

Z

 |u(x)| 6 c|x − x0 |1−n/q 

 |Du|q dy 

B(x0 ,2R)



1/q

Z

 6 cR1−α/q Rα−n

 |Du|q dy 

B(x,4R)

¡ ¢1/q 6 c dist(x, ∂Ω)1−α/q Mα,4δ(x) |Du|q (x)

(5.1)

for every x ∈ Rn with c = c(n, q). This is a pointwise Hardy inequality. For u ∈ W01,p (Ω) this inequality holds almost everywhere. Integrating (5.1) with α = 0 over Ω and using the maximal function theorem, we arrive at ¶p Z Z µ ¡ ¢p/q |u(x)| dx 6 c M |Du|q (x) dx dist(x, ∂Ω) Ω

Ω

Z |Du(x)|p dx

6c

(5.2)

Ω

for every u ∈ W01,p (Ω) with c = c(n, p, q). This is a version of the Hardy inequality which is valid for every open sets with nonempty complement if n < p < ∞. The case 1 < p 6 n is more involved since then extra conditions must be imposed on Ω (see [51, Theorem 3]). However, there is a sufficient condition in terms of capacity density of the complement. A closed set E ⊂ Rn is uniformly p−fat, 1 < p < ∞, if there is a constant γ > 0 such that ¡ ¢ ¡ ¢ capp E ∩ B(x, r), B(x, 2r) > γ capp B(x, r), B(x, 2r) (5.3) for all x ∈ E and r > 0. Here capp (K, Ω) denotes the variational p−capacity Z capp (K, Ω) = inf |Du(x)|p dx, Ω

where the infimum is taken over all u ∈ C0∞ (Ω) such that u(x) > 1 for every x ∈ K. Here Ω is an open subset of Ω and K is a compact subset of Ω. We recall that capp (B(x, r), B(x, 2r)) = crn−p ,

Maximal Functions in Sobolev Spaces

51

where c = c(n, p). If p > n, then all nonempty closed sets are uniformly p-fat. If there is a constant γ > 0 such that E satisfies the measure thickness condition |B(x, r) ∩ E| > γ|B(x, r)| for all x ∈ E and r > 0, then E is uniformly p-fat for every p with 1 < p < ∞. If E is uniformly p-fat for some p, then it is uniformly q-fat for every q > p. The fundamental property of uniformly fat sets is the following self improving result due to Lewis [51, Theorem 1]. For another proof see [61, Theorem 8.2]. Theorem 5.4. Let E ⊂ Rn be a closed uniformly p−fat set. Then there is 1 < q < p such that E is uniformly q−fat. In the case where Ω ⊂ Rn is an open set such that Rn \ Ω is uniformly p−fat, Lewis [51, Theorem 2] proved that the Hardy inequality holds. We have already seen that the Hardy inequality follows from pointwise inequalities involving the Hardy–Littlewood maximal function if p > n. We show that this is also the case 1 < p 6 n. Theorem 5.5. Let 1 < p 6 n, 0 6 α < p, and let Ω ⊂ Rn be an open set such that Rn \ Ω is uniformly p−fat. Suppose that u ∈ C0∞ (Ω). Then there are constants c = c(n, p, γ) and σ > 1 such that ¡ ¢1/p |u(x)| 6 c dist(x, ∂Ω)1−α/p Mα,σδ(x) |Du|p (x)

(5.6)

for every x ∈ Ω. Proof. Let x ∈ Ω. Choose x0 ∈ ∂Ω such that |x − x0 | = dist(x, ∂Ω) = δ(x) = R. Then

¡ ¢1/p |u(x) − uB(x0 ,2R) | 6 cR1−α/p Mα,R |Du|p (x)

for every x ∈ B(x0 , 2R) with c = c(n, p), and hence |u(x)| 6 |u(x) − uB(x0 ,2R) | + |uB(x0 ,2R) | ¡ ¢1/p 6 cR1−α/p Mα,δ(x) |Du|p (x) + |u|B(x0 ,2R) for every x ∈ B(x0 , 2R). Denote A = {x ∈ Rn : u(x) = 0}. Using a capacitary version of the Poincar´e inequality, we arrive at 1 |B(x0 , 2R)|

Z |u| dy B(x0 ,2R)

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Daniel Aalto and Juha Kinnunen

 ¡ ¢−1  6 c capp A ∩ B(x0 , 2R), B(x0 , 4R)

1/p

Z

 |Du|p dy 

B(x0 ,4R)

¡

6 c capp

¡

¢−1 (R \ Ω) ∩ B(x0 , 2R), B(x0 , 4R)

Z

n

|Du|p dy

´1/p

B(x0 ,4R)

  6 c Rp−n

Z

1/p  |Du|p dy 

¡ ¢1/p 6 cR1−α/p Mα,8δ(x) |Du|p (x) ,

B(x,8R)

where c = c(n, p, γ).

u t

If Rn \Ω is p−fat, then, by Theorem 5.4, it is q−fat for some 1 < q < p 6 n. Using (5.6) with α = 0, we get the pointwise q-Hardy inequality ¡ ¢1/q |u(x)| 6 c dist(x, ∂Ω) Mσδ(x) |Du|q (x) for every x ∈ Ω with c = c(n, q). Integrating and using the maximal function theorem exactly in the same way as in (5.2), we also prove the Hardy inequality in the case 1 < p 6 n. Again, a density argument shows that the Hardy inequality holds for every u ∈ W01,p (Ω). Thus, we have proved the following assertion. Corollary 5.7. Let 1 < p < ∞. Suppose that Ω ⊂ Rn is an open set such that Rn \ Ω is uniformly p−fat. If u ∈ W01,p (Ω), then there is a constant c = c(n, p, γ) such that ¶p Z µ Z |u(x)| dx 6 c |Du(x)|p dx. dist(x, ∂Ω) Ω

Ω

In particular, if p > n, then the inequality holds for every Ω 6= Rn . Remark 5.8. The pointwise Hardy inequality is not equivalent to the Hardy inequality since there are open sets for which the Hardy inequality holds for some p, but the pointwise Hardy inequality fails. For example, the punctured ball B(0, 1) \ {0} satisfies the pointwise Hardy inequality only in the case p > n, but the usual Hardy inequality also holds when 1 < p < n. When p = n, the Hardy inequality fails for this set. This example also shows that the uniform fatness of the complement is not a necessary condition for an open set to satisfy the Hardy inequality since the complement of B(0, 1) \ {0} is not uniformly p-fat when 1 < p < n. If p = n, then the Hardy inequality is equivalent to the fact that Rn \ Ω is uniformly p-fat (see [51, Theorem 3]). A recent result of Lehrb¨ack [47] shows that the uniform fatness is not only sufficient, but also necessary condition for the pointwise Hardy inequality

Maximal Functions in Sobolev Spaces

53

(see also [50, 48, 49]). When n = p = 2, Sugawa [65] proved that the Hardy inequality is also equivalent to the uniform perfectness of the complement of the domain. Recently this result was generalized in [43] for other values of p. The arguments of [43] are very general. It is also possible to study Hardy inequalities on metric measure spaces (see [9, 32, 43]). Theorem 5.4 shows that the p-fatness is a self improving result. Next we give a proof of an elegant result of Koskela and Zhong [45] which states that the Hardy inequality is self improving. Theorem 5.9. Suppose that the Hardy inequality holds in Ω for some 1 < p < ∞. Then there exists ε > 0 such that the Hardy inequality holds in Ω for every q with p − ε < q 6 p. Proof. Let u be a Lipschitz continuous function that vanishes in Rn \ Ω. For λ > 0 denote Fλ = {x ∈ Ω : |u(x)| 6 λ dist(x, ∂Ω) and M |Du|(x) 6 λ}. We claim that the restriction of u to Fλ ∪ (Rn \ Ω) is Lipschitz continuous with a constant cλ, where c = c(n). If x, y ∈ Fλ , then |u(x) − u(y)| 6 c|x − y|(M |Du|(x) + M |Du|(y)) 6 cλ|x − y| by Corollary 4.3. If x ∈ Fλ and y ∈ Rn \ Ω, then |u(x) − u(y)| = |u(x)| 6 λ dist(x, ∂Ω) 6 λ|x − y|. This implies that u|Fλ ∪(Rn \Ω) is Lipschitz continuous with the constant cλ. We extend the function to the entire space Rn , for example, with the classical McShane extension v(x) = inf{u(y) + cλ|x − y| : y ∈ Fλ ∪ (Rn \ Ω)}. The function v is Lipschitz continuous in Rn with the same constant cλ as u|Fλ ∪(Rn \Ω) . Let Gλ = {x ∈ Ω : |u(x)| 6 λ dist(x, ∂Ω)} and Eλ = {x ∈ Ω : M |Du|(x) 6 λ}. Then Fλ = Gλ ∩ Eλ , and we note that |Dv(x)| 6 |Du(x)|χFλ (x) + cλχΩ\Fλ (x) for almost all x ∈ Rn . By the Hardy inequality,

54

Daniel Aalto and Juha Kinnunen

Z µ

|v(x)| dist(x, ∂Ω)

¶p

Z |Du(x)|p dx + cλp |Ω \ Fλ |

dx 6 c

Fλ

Fλ

and, consequently, ¶p Z µ |u(x)| dx dist(x, ∂Ω) Gλ

Z

Z p

6c

p

|Du(x)| dx + cλ |Ω \ Fλ | +

Fλ

µ

|u(x)| dist(x, ∂Ω)

¶p dx

Gλ \Eλ

Z |Du(x)|p dx + cλp (|Ω \ Gλ | + |Ω \ Eλ |).

6c Eλ

From this we obtain 1 ε

Z µ

|u(x)| dist(x, ∂Ω)

¶p−ε

6c

λ−ε−1 0

Ω

Z∞

Z µ

Z∞ dx =

−ε−1

0

Gλ

Eλ

Z∞ λ

p−ε−1

λp−ε−1 |Ω \ Eλ | dλ

|Ω \ Gλ | dλ + c 0

0

c ε

Z |Du(x)|p−ε dx +

1 p−ε

Ω

+

dx dλ

|Du(x)|p dx dλ

Z∞

6

¶p

Z λ

+c

|u(x)| dist(x, ∂Ω)

c p−ε

Z µ

|u(x)| dist(x, ∂Ω)

¶p−ε dx

Ω

Z (M |Du|(x))p−ε dx. Ω

The claim follows from this by using the maximal function theorem, choosing ε > 0 small enough, and absorbing the terms on the left-hand side. u t

6 Maximal Functions on Metric Measure Spaces In this section, we show that most of the results that we have discussed so far are based on a general principle and our arguments apply in the context of metric measure spaces.

Maximal Functions in Sobolev Spaces

55

6.1 Sobolev spaces on metric measure spaces Let X = (X, d, µ) be a complete metric space endowed with a metric d and a Borel regular measure µ such that 0 < µ(B(x, r)) < ∞ for all open balls B(x, r) = {y ∈ X : d(y, x) < r} with r > 0. The measure µ is said to be doubling if there exists a constant cµ > 1, called the doubling constant of µ, such that µ(B(x, 2r)) 6 cµ µ(B(x, r)) for all x ∈ X and r > 0. Note that an iteration of the doubling property implies that, if B(x, R) is a ball in X, y ∈ B(x, R), and 0 < r 6 R < ∞, then ³ r ´Q µ(B(y, r)) >c (6.1) µ(B(x, R)) R for some c = c(cµ ) and Q = log cµ / log 2. The exponent Q serves as a counterpart of dimension related to the measure. A nonnegative Borel function g on X is said to be an upper gradient of a function u : X → [−∞, ∞] if for all rectifiable paths γ joining points x and y in X we have Z |u(x) − u(y)| 6

g ds, γ

whenever both u(x) and u(y) are finite, and

R

(6.2) g ds = ∞ otherwise. The

γ

assumption that g is a Borel function is needed in the definition of the path integral. If g is merely a µ-measurable function and (6.2) holds for p-almost every path (i.e., it fails only for a path family with zero p-modulus), then g is said to be a p-weak upper gradient of u. If we redefine a p-weak upper gradient on a set of measure zero we obtain a p-weak upper gradient of the same function. In particular, this implies that, after a possible redefinition on a set of measure zero, we obtain a Borel function. If g is a p-weak upper gradient of u, then there is a sequence gi , i = 1, 2, . . . , of the upper gradients of u such that Z |gi − g|p dµ → 0 X

as i → ∞. Hence every p-weak upper gradient can be approximated by upper gradients in the Lp (X)-norm. If u has an upper gradient that belongs to Lp (X), then it has a minimal p-weak upper gradient gu in the sense that for every p-weak upper gradient g of u, gu 6 g µ-almost everywhere. We define Sobolev spaces on the metric space X using the p-weak upper gradients. For u ∈ Lp (X) we set

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Daniel Aalto and Juha Kinnunen

³Z kukN 1,p (X) =

Z |u|p dµ + inf

g p dµ

g

X

´1/p

,

X

where the infimum is taken over all p-weak upper gradients of u. The Sobolev space (sometimes called the Newtonian space) on X is the quotient space N 1,p (X) = {u : kukN 1,p (X) < ∞}/∼, where u ∼ v if and only if ku − vkN 1,p (X) = 0. The notion of a p-weak upper gradient is used to prove that N 1,p (X) is a Banach space. For properties of Sobolev spaces on metric measure spaces we refer to [30, 29, 62, 63, 6]. The p-capacity of a set E ⊂ X is the number capp (E) = inf kukpN 1,p (X) , where the infimum is taken over all u ∈ N 1,p (X) such that u = 1 on E [38]. We say that a property regarding points in X holds p-quasieverywhere (p-q.e.) if the set of points for which the property does not hold has capacity zero. If u ∈ N 1,p (X), then u ∼ v if and only if u = v p-q.e. Moreover, if u, v ∈ N 1,p (X) and u = v µ-a.e., then u ∼ v. Hence the capacity is the correct gauge for distinguishing between two Newtonian functions (see [8]). To be able to compare the boundary values of Sobolev functions, we need a Sobolev space with zero boundary values. Let E be a measurable subset of X. The Sobolev space with zero boundary values is the space N01,p (E) = {u|E : u ∈ N 1,p (X) and u = 0 p-q.e. in X \ E}. The space N01,p (E) equipped with the norm inherited from N 1,p (X) is a Banach space. We say that X supports a weak (1, p)-Poincar´e inequality if there exist constants c > 0 and λ > 1 such that for all balls B(x, r) ⊂ X, all locally integrable functions u on X and for all p-weak upper gradients g of u, Z ³ Z ´1/p |u − uB(x,r) | dµ 6 cr g p dµ , (6.3) B(x,r)

where

B(x,λr)

Z uB(x,r) = B(x,r)

1 u dµ = µ(B(x, r))

Z u dµ. B(x,r)

Since the p-weak upper gradients can be approximated by upper gradients in the Lp (X)-norm, we could require the Poincar´e inequality for upper gradients as well. By the H¨older inequality, it is easy to see that if X supports a weak (1, p)Poincar´e inequality, then it supports a weak (1, q)-Poincar´e inequality for

Maximal Functions in Sobolev Spaces

57

every q > p. If X is complete and µ doubling then it is shown in [31] that a weak (1, p)-Poincar´e inequality implies a weak (1, q)-Poincar´e inequality for some q < p. Hence the (1, p)-Poincar´e inequality is a self improving condition. For simplicity, we assume throughout that X supports a weak (1, 1)-Poincar´e inequality, although, by using the results of [31], it would be enough to assume that X supports a weak (1, p)-Poincar´e inequality. We leave the extensions to the interested reader. In addition, we assume that X is complete and µ is doubling. This implies, for example, that Lipschitz functions are dense in N 1,p (X) and the Sobolev embedding theorem holds.

6.2 Maximal function defined on the whole space The standard centered Hardy–Littlewood maximal function on a metric measure space X is defined as Z M u(x) = sup |u| dµ. r>0 B(x,r)

By the Hardy–Littlewood maximal function theorem for doubling measures (see [15]), we see that the Hardy–Littlewood maximal operator is bounded on Lp (X) when 1 < p 6 ∞ and maps L1 (X) into the weak L1 (X). However, the standard Hardy–Littlewood maximal function does not seem to preserve the smoothness of the functions as examples by Buckley [12] clearly indicate. In order to have a maximal function which preserves, for example, the Sobolev spaces on metric measure spaces, we construct a maximal function based on a discrete convolution. Let r > 0. We begin by constructing a family of balls which cover the space and are of bounded overlap. Indeed, there is a family of balls B(xi , r), i = 1, 2, . . . , such that ∞ [ X= B(xi , r) i=1

and

∞ X

χB(xi ,6r) 6 c < ∞.

i=1

This means that the dilated balls B(xi , 6r) are of bounded overlap. The constant c depends only on the doubling constant and, in particular, is independent of r. These balls play the role of Whitney cubes in a metric measure space. Then we construct a partition of unity subordinate to the cover B(xi , r), i = 1, 2, . . . , of X. Indeed, there is a family of functions ϕi , i = 1, 2, . . . , such that 0 6 ϕi 6 1, ϕi = 0 on X \ B(xi , 6r), ϕi > c on B(xi , 3r), ϕi is Lipschitz

58

Daniel Aalto and Juha Kinnunen

with constant c/ri with c depending only on the doubling constant, and ∞ X

ϕi = 1

i=1

in X. The partition of unity can be constructed by first choosing auxiliary cutoff functions ϕ ei so that 0 6 ϕ ei 6 1, ϕ ei = 0 on X \ B(xi , 6r), ϕ ei = 1 on B(xi , 3r) and each ϕ ei is Lipschitz with constant c/r. We can, for example, take   1, x ∈ B(xi , 3r),   d(x, xi ) ϕ ei (x) = 2 − , x ∈ B(xi , 6r) \ B(xi , 3r),  3r   0, x ∈ X \ B(xi , 6r). Then we can define the functions ϕi , i = 1, 2, . . . , in the partition of unity by ϕ ei (x) ϕi (x) = P . ∞ ϕ ej (x) j=1

It is not difficult to see that the defined functions satisfy the required properties. Now we are ready to define the approximation of u at the scale of 3r by setting ∞ X ur (x) = ϕi (x)uB(xi ,3r) i=1

for every x ∈ X. The function ur is called the discrete convolution of u. The partition of unity and the discrete convolution are standard tools in harmonic analysis on homogeneous spaces (see, for example, [15] and [57]). Let rj , j = 1, 2, . . . , be an enumeration of the positive rational numbers. For every radius rj we choose balls B(xi , rj ), i = 1, 2, . . . , of X as above. Observe that for each radius there are many possible choices for the covering, but we simply take one of those. We define the discrete maximal function related to the coverings B(xi , rj ), i, j = 1, 2, . . . , by M ∗ u(x) = sup |u|rj (x) j

for every x ∈ X. We emphasize the fact that the defined maximal operator depends on the chosen coverings. This is not a serious matter since we obtain estimates which are independent of the chosen coverings. As the supremum of continuous functions, the discrete maximal function is lower semicontinuous and hence measurable. The following result shows that the discrete maximal function is equivalent with two-sided estimates to the standard Hardy–Littlewood maximal function.

Maximal Functions in Sobolev Spaces

59

Lemma 6.4. There is a constant c > 1, which depends only on the doubling constant, such that c−1 M u(x) 6 M ∗ u(x) 6 cM u(x) for every x ∈ X. Proof. We begin by proving the second inequality. Let x ∈ X, and let rj be a positive rational number. Since ϕi = 0 on X \ B(xi , 6rj ) and B(xi , 3rj ) ⊂ B(x, 9rj ) for every x ∈ B(xi , 6rj ), we have, by the doubling condition, |u|rj (x) =

∞ X

ϕi (x)|u|B(xi ,3rj )

i=1

6

∞ X i=1

ϕi (x)

µ(B(x, 9rj )) µ(B(xi , 3rj ))

Z |u| dµ 6 cM u(x), B(x,9rj )

where c depends only on the doubling constant cµ . The second inequality follows by taking the supremum on the left-hand side. To prove the first inequality, we observe that for each x ∈ X there exists i = ix such that x ∈ B(xi , rj ). This implies that B(x, rj ) ⊂ B(xi , 2rj ) and hence Z Z |u| dµ 6 c |u| dµ B(x,rj )

B(xi ,3rj )

Z |u| dµ 6 cM ∗ u(x).

6 cϕi (x) B(xi ,3rj )

In the second inequality, we used the fact that ϕi > c on B(xi , rj ). Again, the claim follows by taking the supremum on the left-hand side. u t Since the maximal operators are comparable, we conclude that the maximal function theorem holds for the discrete maximal operator as well. Our goal is to show that the operator M ∗ preserves the smoothness of the function in the sense that it is a bounded operator in N 1,p (X). We begin by proving the corresponding result for the discrete convolution in a fixed scale. Lemma 6.5. Suppose that u ∈ N 1,p (X) with p > 1. Let r > 0. Then |u|r ∈ N 1,p (X) and there is a constant c, which depends only on the doubling constant, such that cM ∗ gu is a p-weak upper gradient of |u|r whenever gu is a p-weak upper gradient of u. Proof. By Lemma 6.4, we have |u|r 6 cM u. By the maximal function theorem with p > 1, we conclude that |u|r ∈ Lp (X).

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Then we consider the upper gradient. We have |u|r (x) =

∞ X

ϕi (x)|u|B(xi ,3r)

i=1

= |u(x)| +

∞ X

¡ ¢ ϕi (x) |u|B(xi ,3r) − |u(x)| .

i=1

Observe that, at each point, the sum is taken only over finitely many balls so that the convergence of the series is clear. Let g|u| be a p-weak upper gradient of |u|. Then ∞ X g|u| + gϕi (|u|B(xi ,3r) −|u|) i=1

is a p-weak upper gradient of |u|r . On the other hand, ³c¯ ´ ¯ ¯|u| − |u|B(x ,3r) ¯ + g|u| χB(x ,6r) i i r is a p-weak upper gradient of ϕi (|u|B(xi ,3r) − |u|). Let gr = gu +

∞ ³ ´ X ¯ c ¯¯ |u| − |u|B(xi ,3r) ¯ + gu χB(xi ,6r) . r i=1

Then gr is a p-weak upper gradient of |u|r . Here we used the fact that every p-weak upper gradient of u will do as a p-weak upper gradient of |u| as well. Then we show that gr ∈ Lp (X). Let x ∈ B(xi , 6r). Then B(xi , 3r) ⊂ B(x, 9r) and ¯ ¯ ¯ ¯ ¯ ¯ ¯|u(x)| − |u|B(x ,3r) ¯ 6 ¯|u(x)| − |u|B(x,9r) ¯ + ¯|u|B(x,9r) − |u|B(x ,3r) ¯. i i We estimate the second term on the right-hand side by the Poincar´e inequality and the doubling condition as Z ¯ ¯ ¯ ¯ ¯|u| − |u|B(x,9r) ¯ dµ ¯|u|B(x,9r) − |u|B(x ,3r) ¯ 6 i Z 6c

B(xi ,3r)

¯ ¯ ¯|u| − |u|B(x,9r) ¯ dµ 6 cr

B(x,9r)

Z gu dµ. B(x,9r)

The first term on the right-hand side is estimated by a standard telescoping argument. Since µ-almost every point is a Lebesgue point for u, we have ∞ ¯ ¯ X ¯ ¯ ¯|u(x)| − |u|B(x,9r) ¯ 6 ¯|u|B(x,32−j r) − |u|B(x,31−j r) ¯ j=0

Maximal Functions in Sobolev Spaces

6c

∞ X j=0

6c

∞ X j=0

Z

61

¯ ¯ ¯|u| − |u|B(x,32−j r) ¯ dµ

B(x,32−j r)

Z 32−j r

gu dµ 6 crM gu (x) B(x,32−j r)

for µ-almost all x ∈ X. Here we used the Poincar´e inequality and the doubling condition again. Hence we have Z ¯ ¯ ¯|u(x)| − |u|B(x ,3r) ¯ 6 cr gu dµ + crM gu (x) 6 crM gu (x) i B(x,9r)

for µ-almost all x ∈ X. From this we conclude that gr = gu +

∞ ³ ´ X ¯ c ¯¯ |u| − |u|B(xi ,3r) ¯ + gu χB(xi ,6r) 6 cM gu (x) r i=1

for µ-almost all x ∈ X. Here c depends only on the doubling constant. This implies that cM gu is a p-weak upper gradient of ur . The maximal function theorem shows that gr ∈ Lp (X) since p > 1. u t Now we are ready to conclude that the discrete maximal operator preserves Newtonian spaces. We use the following simple fact in the proof. Suppose that ui , i = 1, 2, . . . , are functions and gi , i = 1, 2, . . . , are p-weak upper gradients of ui respectively. Let u = supi ui , and let g = supi gi . If u < ∞ µ−almost everywhere, then g is a p-weak upper gradient of u. For the proof, we refer to [6]. The following result is a counterpart of Theorem 2.2 in metric measure spaces. Theorem 6.6. If u ∈ N 1,p (X) with p > 1, then M ∗ u ∈ N 1,p (X). In addition, the function cM ∗ gu is a p-weak upper gradient of M ∗ u whenever gu is a p-weak upper gradient of u. The constant c depends only on the doubling constant. Proof. By the maximal function theorem, we see that M ∗ u ∈ Lp (X) and, in particular, M ∗ u < ∞ µ-almost everywhere. Since M ∗ u(x) = sup |u|rj (x) j

and cM ∗ gu is an upper gradient of |u|rj for every j, we conclude that it is an upper gradient of M ∗ u as well. The claim follows from the maximal function theorem. u t Remark 6.7. (i) By Theorem 6.6 and the Hardy–Littlewood maximal theorem, we conclude that the discrete maximal operator M ∗ is bounded in N 1,p (X).

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(ii) The fact that the maximal operator is bounded in N 1,p (X) can be used to prove a capacitary weak type estimate in metric spaces. This implies that u ∈ N 1,p (X) has Lebesgue points outside a set of p-capacity zero (see [7] and [34]).

6.3 Maximal function defined on a subdomain This subsection is based on [1]. We recall the following Whitney type covering theorem (see [15] and [57]). Lemma 6.8. Let Ω ⊂ X be an open set with a nonempty complement. Then for every 0 < t < 1 there are balls B(xi , ri ) ⊂ Ω, i = 1, 2, . . . , such that ∞ [

B(xi , ri ) = Ω,

i=1

for every x ∈ B(xi , 6ri ), i = 1, 2, . . . , we have c1 ri 6 t dist(x, X \ Ω) 6 c2 ri and the balls B(xi , 6ri ), i = 1, 2, . . . , are of bounded overlap. Here the constants c1 and c2 depend only on the doubling constant. In particular, the bound for the overlap is independent of the scale t. Let 0 < t < 1 be a rational number. We consider a Whitney type decomposition of Ω. We construct a partition of unity and discrete convolution related to the Whitney balls exactly in the same way as before. Let tj , j = 1, 2, . . . , be an enumeration of the positive rational numbers. of the interval (0, 1). For every scale tj we choose a Whitney covering as in Lemma 6.8 and construct a discrete convolution |u|tj . Observe that for each scale there are many possible choices for the covering, but we simply take one of those. We define the discrete maximal function related to the discrete convolution |u|tj by ∗ MΩ u(x) = supj |u|tj (x)

for every x ∈ X. Again, the defined maximal operator depends on the chosen coverings, but this is not a serious matter for the same reason as above. It can be shown that there is a constant c > 1, depending only on the doubling constant, such that ∗ MΩ u(x) 6 cMΩ u(x) for every x ∈ Ω. Here, Z |u| dµ MΩ u(x) = sup B(x,r)

Maximal Functions in Sobolev Spaces

63

is the standard maximal function related to the open subset Ω ⊂ X and the supremum is taken over all balls B(x, r) contained in Ω. There is also an inequality to the reverse direction, but then we have to restrict ourselves in the definition of the maximal function to such balls that B(x, σr) is contained in Ω for some σ large enough. The pointwise inequality implies that the ∗ as well. maximal function theorem holds for MΩ Using a similar argument as above, we can show that, if the measure µ is doubling and the space supports a weak (1,1)-Poincar´e inequality, then the ∗ maximal operator MΩ preserves the Sobolev spaces N 1,p (Ω) for every open Ω ⊂ X when p > 1. Moreover, ∗ : N 1,p (Ω) → N 1,p (Ω) MΩ

is a bounded operator when p > 1. It is an interesting open question to study the continuity of the operator and the borderline case p = 1. Then we consider the Sobolev boundary values. The following assertion is a counterpart of Theorem 3.12 in metric measure spaces. Theorem 6.9. Let Ω ⊂ X be an open set. Assume that u ∈ N 1,p (Ω) with p > 1. Then ∗ |u| − MΩ u ∈ N01,p (Ω). Proof. Let 0 < t < 1. Consider the discrete convolution |u|t . Let x ∈ Ω with x ∈ B(xi , ri ). Using the same telescoping argument as in the proof of Lemma 4.1 and the properties of the Whitney balls we have ¯ ¯ ¯|u|B(x ,3r ) − |u(x)|¯ 6 cri MΩ gu (x) 6 ct dist(x, ∂Ω)MΩ gu (x). i

i

It follows that ∞ ¯ ¯ ¯X ¡ ¯|u|t (x) − |u(x)|¯ = ¯¯ ψi (x) |u|B(x

¢¯¯ − |u(x)| ¯ i ,3ri )

i=1

6

∞ X

¯ ¯ ψi (x)¯|u|B(xi ,3ri ) − |u(x)|¯

i=1

6 ct dist(x, ∂Ω)MΩ gu (x). For every x ∈ Ω there is a sequence tj , j = 1, 2, . . . , of scales such that ∗ MΩ u(x) = lim |u|tj (x) j→∞

This implies that ¯ ¯ ∗ ¯|u(x)| − MΩ u(x)¯ = lim ||u(x) − |u|tj (x)|| j→∞

6 c dist(x, ∂Ω)MΩ gu (x),

64

Daniel Aalto and Juha Kinnunen

where we used the fact that tj 6 1. Hence, by the maximal function theorem, we conclude that ¯ !p Z à ¯¯ Z ∗ |u(x)| − MΩ u(x)¯ dµ(x) 6 c (MΩ gu (x))p dµ(x) dist(x, ∂Ω) Ω

Ω

Z |gu (x)|p dµ(x).

6c Ω

This implies that

∗ |u(x)| − MΩ u(x) ∈ Lp (Ω) dist(x, ∂Ω)

∗ and from Theorem 5.1 in [32] we conclude that |u| − MΩ u ∈ N01,p (Ω).

u t

6.4 Pointwise estimates and Lusin type approximation Let u be a locally integrable function in X, let 0 6 α < 1, and let β = 1 − α. From the proof of Lemma 4.1 it follows that ¡ ¢ # |u(x) − u(y)| 6 c d(x, y)β u# β,4d(x,y) (x) + uβ,4d(x,y) (y) for every x 6= y. By the weak Poincar´e inequality, u# β,4d(x,y) (x) 6 cMα,4λd(x,y) gu (x) for every x ∈ X. Denote Eλ = {x ∈ X : Mα gu (x) > λ}, where λ > 0. We see that u|X\Eλ is H¨older continuous with the exponent β. We can extend this function to a H¨older continuous function on X by using a Whitney type extension. The Whitney type covering lemma (Lemma 6.8) enables us to construct a partition of unity as above. Let B(xi , ri ), i = 1, 2, . . . , be the Whitney covering of the open set Eλ . Then there are nonnegative functions ϕi , i = 1, 2, . . ., such that ϕi = 0 in X \ B(xi , 6ri ), 0 6 ϕi (x) 6 1 for every x ∈ X, every ϕi is Lipschitz with the constant c/ri and ∞ X ϕi (x) = χEλ (x) i=1

for every x ∈ X. We define the Whitney smoothing of u by

Maximal Functions in Sobolev Spaces

uλ (x) =

65

  u(x),

x ∈ X \ Eλ ,

∞ P   ϕi (x)uB(xi ,3ri ) ,

x ∈ Eλ .

i=1

We obtain the following result by similar arguments as above. The exponent Q refers to the dimension given by (6.1). Theorem 6.10. Suppose that u ∈ N 1,p (X), 1 < p 6 Q. Let 0 6 α < 1. Then for every λ > 0 there is a function uλ and an open set Eλ such that u = uλ everywhere in X \ Eλ , uλ ∈ N 1,p (X), and uλ is H¨ older continuous with the exponent 1 − α on every bounded set in X, ku − uλ kN 1,p (X) → 0, n−αp and H∞ (Eλ ) → 0 as λ → ∞.

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